Countable Image Research Articles

Comparing functional countability and exponential separability

A space is functionally countable if every real-valued continuous function has countable image. A stronger property recently defined by Tkachuk is exponential separability. We start by studying these properties in GO spaces, where we extend results by Tkachuk and Wilson, and prove a conjecture by Dow. We also study some subspaces of products that are functionally countable and the influence of the Gδ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_\\delta $$\\end{document}-topology on exponential separability. Finally, we give some examples of functionally countable spaces that are separable and uncountable.

On CP-frames

Let $mathcal{R}_c( L)$ be the pointfree version of $C_c(X)$, the subring of $C(X)$ whose elements have countable image. We shall call a frame $L $ a $CP$-frame if thering $mathcal{R}_c( L)$ is regular. % The main aim of this paper is to introduce $CP$-frames, that is $mathcal{R}_c( L)$ is a regular ring. We give some We give some characterizations of $CP$-frames and we show that $L$ is a $CP$-frame if and only if each prime ideal of $mathcal{R}_c ( L)$ is an intersection of maximal ideals if and only if every ideal of $mathcal{R}_c ( L)$ is a $z_c$-ideal. In particular, we prove that any $P$-frame is a $CP$-frame but not conversely, in general. In addition, we study some results about $CP$-frames like the relation between a $CP$-frame $L$ and ideals of closed quotients of $L$. Next, we characterize $CP$-frames as precisely those $L$ for which every prime ideal in the ring $mathcal{R}_c ( L)$ is a $z_c$-ideal. Finally, we show that this characterization still holds if prime ideals are replaced by essential ideals, radical ideals, convex ideals, or absolutely convex ideals.

Locally functionally countable subalgebra of $\mathcal{R}(L)$

Let $L_c(X)= \lbrace f \in C(X) \colon \overline{C_f}= X\rbrace $, where $C_f$ is the union of all open subsets $U \subseteq X$ such that $\vert f(U) \vert \le \aleph _0$. In this paper, we present a pointfree topology version of $L_c(X)$, named $\mathcal{R}_{\ell c}(L)$. We observe that $\mathcal{R}_{\ell c}(L)$ enjoys most of the important properties shared by $\mathcal{R}(L)$ and $\mathcal{R}_c(L)$, where $\mathcal{R}_c(L)$ is the pointfree version of all continuous functions of $C(X)$ with countable image. The interrelation between $\mathcal{R}(L)$, $\mathcal{R}_{\ell c}(L)$, and $\mathcal{R}_c(L)$ is examined. We show that $L_c(X) \cong \mathcal{R}_{\ell c}\big (\mathfrak{O}(X)\big )$ for any space $X$. Frames $L$ for which $\mathcal{R}_{\ell c}(L)=\mathcal{R}(L)$ are characterized.

Further thoughts on the ring $${\mathcal {R}}_c (L)$$ in frames

Let C(X) denote the ring of all real-valued continuous functions on a topological space X and $${\mathcal {R}} (L)$$ be as the pointfree topology version of C(X), i.e., the ring of real-valued continuous functions on a frame L. The ring $${\mathcal {R}}_c (L)$$ is introduced as a sub-f-ring of $${\mathcal {R}} ( L)$$ as a pointfree analogue to the subring $$C_c(X)$$ of C(X) consisting of elements with the countable image. In this paper, we will study the concept of pointfree countable image in a way which will enable us to study the ring $${\mathcal {R}}_c (L)$$. In order to do so we introduce the set $$R_{\alpha } := \{ r \in {\mathbb {R}} : {{\,\mathrm{coz}\,}}(\alpha - \mathbf{r}) \not = \top \} $$ for every $$\alpha \in {\mathcal {R}} (L)$$. We prove that $$R_{\alpha } $$ is a countable subset of $${\mathbb {R}}$$ for every $$\alpha \in {\mathcal {R}}_c (L)$$. Next, we show that if L is a compact frame, then $$R_{\alpha } $$ is a finite subset of $${\mathbb {R}}$$ for every $$\alpha \in {\mathcal {R}}_c (L)$$. Also, we study the result which says that for any topological space X there is a zero-dimensional space Y which is a continuous image of X and $$C_c(X) \cong C_c(Y )$$ in pointfree topology. Finally, we prove that, for some frame L, the ring $${\mathcal {R}}_c (L)$$ may not be isomorphic to $${\mathcal {R}} (M)$$, for any given frame M.

When is the super socle of C(X) prime?

<p>Let SC<sub>F</sub>(X) denote the ideal of C(X) consisting of functions which are zero everywhere except on a countable number of points of X. It is generalization of the socle of C(X) denoted by C<sub>F</sub>(X). Using this concept we extend some of the basic results concerning C<sub>F</sub>(X) to SC<sub>F</sub>(X). In particular, we characterize the spaces X such that SC<sub>F</sub>(X) is a prime ideal in C(X) (note, C<sub>F</sub>(X) is never a prime ideal in C(X)). This may be considered as an advantage of SC<sub>F</sub>(X) over C(X). We are also interested in characterizing topological spaces X such that C<sub>c</sub>(X) =R+SC<sub>F</sub>(X), where C<sub>c</sub>(X) denotes the subring of C(X) consisting of functions with countable image.</p>

Pointfree topology version of image of real-valued continuous functions

Let $ { mathcal{R}} L$ be the ring of real-valued continuous functions on a frame $L$ as the pointfree version of $C(X)$, the ring of all real-valued continuous functions on a topological space $X$. Since $C_c(X)$ is the largest subring of $C(X)$ whose elements have countable image, this motivates us to present the pointfree version of $C_c(X).$The main aim of this paper is to present the pointfree version of image of real-valued continuous functions in $ {mathcal{R}} L$. In particular, we will introduce the pointfree version of the ring $C_c(X)$. We define a relation from $ {mathcal{R}} L$ into the power set of $mathbb R$, namely overlap . Fundamental properties of this relation are studied. The relation overlap is a pointfree version of the relation defined as $mathop{hbox{Im}} (f) subseteq S$ for every continuous function $f:Xrightarrowmathbb R$ and $ S subseteq mathbb R$.

Zc-ideals and prime ideals in the ring RcL

The ring RcL is introduced as a sub-f-ring of RL as a pointfree analogue to the subring Cc(X) of C(X) consisting of elements with the countable image. We introduce zc-ideals in RcL and study their properties. We prove that for any frame L, there exists a space X such that ?L ? OX with Cc(X) ? Rc(OX) ? Rc?L ? R*cL, and from this, we conclude that if ?,? ? RcL, |?| ? |?|q for some q > 1, then ? is a multiple of ? in RcL. Also, we show that IJ = I ? J whenever I and J are zc-ideals. In particular, we prove that an ideal of RcL is a zc-ideal if and only if it is a z-ideals. In addition, we study the relation between zc-ideals and prime ideals in RcL. Finally, we prove that RcL is a Gelfand ring.

On the locally countable subalgebra of C(X) whose local domain is cocountable

In this paper, we present a new subring of $C(X)$ that contains the subring $C_c(X)$, the set of all continuous functions with countable image. Let $L_{cc}(X)=\{ f\in C(X)\,:\, |X\backslash C_f|\leq \aleph_0 \}$, where $C_f$ is the union of all open subsets $U\subseteq X$ such that $|f(U)|\leq \aleph_0$. We observe that $L_{cc}(X)$ enjoys most of the important properties which are shared by $C(X)$ and $C_c(X)$. It is shown that any hereditary lindel\"{o}f scattered space is functionally countable.Spaces $X$ such that $L_{cc}(X)$ is regular (von Neumann) are characterized and it is shown that $\aleph_0$-selfinjectivity and regularity of $L_{cc}(X)$ coincide.

Functionally countable subalgebras and some properties of the Banaschewski compactification

Let $X$ be a zero-dimensional space and $C_c(X)$ be the set of all continuous real valued functions on $X$ with countable image. In this article we denote by $C_c^K(X)$ (resp., $C_c^{\psi}(X))$ the set of all functions in $C_c(X)$ with compact (resp., pseudocompact) support. First, we observe that $C_c^K(X)=O_c^{\beta_0X\setminus X}$ (resp., $C^{\psi}_c(X)=M_c^{\beta_0X\setminus \upsilon_0X}$), where $\beta_0X$ is the Banaschewski compactification of $X$ and $\upsilon_0X$ is the $\mathbb{N}$-compactification of $X$. This implies that for an $\mathbb{N}$-compact space $X$, the intersection of all free maximal ideals in $C_c(X)$ is equal to $C_c^K(X)$, i.e., $M_c^{\beta_0X\setminus X}=C_c^K(X)$. By applying methods of functionally countable subalgebras, we then obtain some results in the remainder of the Banaschewski compactification. We show that for a non-pseudocompact zero-dimensional space $X$, the set $\beta_0X\setminus \upsilon_0X$ has cardinality at least $2^{2^{\aleph_0}}$. Moreover, for a locally compact and $\mathbb{N}$-compact space $X$, the remainder $\beta_0X\setminus X$ is an almost $P$-space. These results lead us to find a class of Parovicenko spaces in the Banaschewski compactification of a non pseudocompact zero-dimensional space. We conclude with a theorem which gives a lower bound for the cellularity of the subspaces $\beta_0X\setminus \upsilon_0X$ and $\beta_0X\setminus X$, whenever $X$ is a zero-dimensional, locally compact space which is not pseudocompact.

A note on extreme sets for quasi-continuous functions

It is known that for Ω a connected separable metric space, a function Ω → R which at every point attains a weak local extremum must have countable image. Let Ω ⊂ R n be a bounded domain and f :Ω → C a measurable, bounded, quasi-continuous function. For n ∈ Z+, we show that if |f | has countable image then f attains weak local extremum at each point of a dense open subset. We show by examples the optimality of the result, in the sense that strengthening the countability condition to discreteness (or even finiteness) will not affect the dense open condition. Conversely it is easily verified that if f attains weak local extremum off a countable set then |f | has countable image. Mathematics Subject Classification: 26A15, 54C08, 54C30

Locally countable properties and the perceptual salience of textures

The human ability to discriminate structured from uniformly random binary textures has been shown to exploit third- and higher-order pixel correlations. We examine this ability in an experiment using a large number of texture families that can only be distinguished on the basis of these higher-order correlations. This study investigates statistical models based on possible explanatory variables involving spatial interactions of up to four pixels. Some of these explanatory variables have been recently associated with natural images, and others are somewhat less intuitive and are used here for the first time, to our knowledge. Our models are constructed using intraclass and cross-class feature selection by means of lasso/elastic net optimization and extensive cross-validation. We focus on a special set of locally countable image measures that seem to parsimoniously capture the observed discrimination performance. Among the measures underpinning the best models, we highlight a concept that can only exist in nine-pixel or larger image patches, but nonetheless is calculable based on the multiplicity of specific four-pixel patches in a texture. We show that this single geometric concept provides significant clues to explain texture discrimination.

Endothelial Image Quality After Descemet Stripping With Endothelial Keratoplasty: A Comparison of Three Microscopy Techniques

To determine corneal endothelial image quality after descemet stripping with endothelial keratoplasty (DSEK) as recorded by three different microscopy techniques: noncontact specular microscopy, noncontact confocal microscopy, and contact confocal microscopy. The corneal endothelium of 52 eyes after DSEK and 20 normal eyes was photographed by the three microscopy techniques during a single encounter. Image quality was graded by two masked observers according to the proportion of countable contiguous cells visible in the image; disagreements in grading were adjudicated by a third observer. Endothelial cell density was compared among the three techniques. After DSEK, image quality was better with contact confocal microscopy than with noncontact confocal microscopy (P = 0.01) and better with noncontact confocal microscopy than with noncontact specular microscopy (P < 0.001). With noncontact specular microscopy, 42% of images after DSEK were uncountable. In normal corneas, all images were countable, and although image quality was better with contact confocal microscopy than with noncontact confocal (P = 0.03) and noncontact specular (P < 0.001) microscopy, the difference was not clinically important. For countable images, the mean differences in endothelial cell density between microscopy methods were close to zero after DSEK and in normal corneas. Confocal optics enable better image quality of the corneal endothelium in corneas with high backscatter, such as those after DSEK. When images were countable, there was a good agreement for endothelial cell density measured by the three microscopy techniques.

Projective versions of selection principles

All spaces are assumed to be Tychonoff. A space X is called projectively P (where P is a topological property) if every continuous second countable image of X is P . Characterizations of projectively Menger spaces X in terms of continuous mappings f : X → R ω , of Menger base property with respect to separable pseudometrics and a selection principle restricted to countable covers by cozero sets are given. If all finite powers of X are projectively Menger, then all countable subspaces of C p ( X ) have countable fan tightness. The class of projectively Menger spaces contains all Menger spaces as well as all σ-pseudocompact spaces, and all spaces of cardinality less than d . Projective versions of Hurewicz, Rothberger and other selection principles satisfy properties similar to the properties of projectively Menger spaces, as well as some specific properties. Thus, X is projectively Hurewicz iff C p ( X ) has the Monotonic Sequence Selection Property in the sense of Scheepers; βX is Rothberger iff X is pseudocompact and projectively Rothberger. Embeddability of the countable fan space V ω into C p ( X ) or C p ( X , 2 ) is characterized in terms of projective properties of X.

Modular actions and amenable representations

Consider a measure-preserving action Γ ↷ (X, μ) of a countable group Γ and a measurable cocycle α: X × Γ → Aut(Y) with countable image, where (X, μ) is a standard Lebesgue space and (Y, ν) is any probability space. We prove that if the Koopman representation associated to the action Γ ↷ X is non-amenable, then there does not exist a countable-to-one Borel homomorphism from the orbit equivalence relation of the skew product action Γ ↷^α X × Y to the orbit equivalence relation of any modular action (i.e., an inverse limit of actions on countable sets or, equivalently, an action on the boundary of a countably-splitting tree), generalizing previous results of Hjorth and Kechris. As an application, for certain groups, we connect antimodularity to mixing conditions. We also show that any countable, non-amenable, residually finite group induces at least three mutually orbit inequivalent free, measure-preserving, ergodic actions as well as two non-Borel bireducible ones.

The Steel hierarchy of ordinal valued Borel mappings

AbstractGiven well ordered countable sets of the form Λϕ, we consider Borel mappings from Λϕω with countable image inside the ordinals. The ordinals and Λϕω are respectively equipped with the discrete topology and the product of the discrete topology on Λϕ. The Steel well-ordering on such mappings is denned by ϕ ≤ ψ iff there exists a continuous function f such that ϕ(x) ≤ ψof(x) holds for any x ϵ Λϕω. It induces a hierarchy of mappings which we give a complete description of. We provide, for each ordinal α, a mapping whose rank is precisely α in this hierarchy and we also compute the height of the hierarchy restricted to mappings with image bounded by α. These mappings being viewed as partitions of the reals, there is, in most cases, a unique distinguished element of the partition. We analyze the relation between its topological complexity and the rank of the mapping in this hierarchy.

Closed mappings and quasimetrics

Closed continuous mappings with first countable images preserve quasi-metric spaces as well as nonarchimedean quasi-metric spaces and γ \gamma -spaces. This is a strict generalization of analogous results on perfect mappings: there exists a closed continuous mapping of a nonarchimedean quasi-metric Moore space onto a compact metric space which is neither perfect nor boundary compact.